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Distributional and inferential properties of theprocess accuracy and process precision indicesW.L. Pearn a , G.H. Lin b & K.S. Chen ca Department of Industrial Engineering & Management, National Chiao Tung University,Hsinchu, Taiwanb Department of Industrial Engineering & Management, National Chiao Tung University,Hsinchu, Taiwanc Department of Industrial Engineering & Management, National Chin-Yi Institute ofTechnology, Taichung, TaiwanPublished online: 27 Jun 2007.

To cite this article: W.L. Pearn , G.H. Lin & K.S. Chen (1998) Distributional and inferential properties of the processaccuracy and process precision indices, Communications in Statistics - Theory and Methods, 27:4, 985-1000, DOI:10.1080/03610929808832139

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COMMUN. STATIST.-THEORY METH., 27(4), 985-1000 (1998)

DISTRIBUTIONAL AND INFERENTIAL PROPERTIES OF THE PROCESS ACCURACY AND

PROCESS PRECISION INDICES

W. L. PEARN Department of Industrial Engineering & Management

National Chiao Tung University Hsinchu. Taiwan ROC

G. H. LIN Department of Industrial Engineering & Management

National Chiao Tung University Hsinchu. Taiwan ROC

K. S . CHEN Department of Industrial Engineering & Management

National Chin-Yi Institute of Technology Taichung, Taiwan ROC

Keywords a~td Phrases: process accuracy index; process precision index; process yield; process mean; process standard deviation.

ABSTRACT

Process capability indices such as C,, k, and Cpk, have been widely used in manufacturing industry to provide numerical measures on process potential and performance. While C, measures overall process variation, k measures the degree of process departure. In this paper, we consider the index C, and a transformation of k defined as C, = 1 - k which measures the degree of process centering. W e refer to C, as the process precision index, and C, as the process accuracy index. W e consider the estimators of C, and Ca, and investigate their statistical

Copyright O 1998 by Marcel Dekker. Inc.

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986 PEARN, LIN, AND CHEN

properties. For C,, we obtain the UMVUE and the MLE. We show that this UMVUE is consistent, and asymptotically efficient. For C,, we investigate its natural estimator. W e obtain the first two moments of this estimator, and show that the natural estimator is the MLE, which is asymptotically unbiased and asymptotically efficient. We also propose an efficient test based on the UMVUE of C,. W e show that the proposed test is the UMP test.

1. INTRODUCTION

Process capability indices, which establish the relationships between the actual process performance and the manufacturing specifications (including the target value and specification limits), have been the focus of recent research in quality assurance and process capability (quality) analysis. Those capability indices, quantifying process potential and performance, are important for any successful quality improvement activities and quality program implementation. Several indices widely used in manufacturing industry providing numerical measures on whether a process meets the preset quality requirement, include C,, k, and Cpk which are defined as the following (see Kane (1986)):

c, = YSL - LSL 60 '

U S L - p p - LSL C* = min (--- -- I 3 0 ' 30 '

where USL and LSL are the upper and the lower specificat~on limits preset by the process engineers or product designers, p is the process mean, u is the process standard deviation, m is the mid-point between the upper and the lower specification limits (m = (USL + LSL)/2), and d is half length of the specification interval (d = (USL - LSL)/2). We have assumed the target value T = m (which is quite common in practical situarions) for simplicity of our discussions.

The index C, was designed to measure the magnitude of the overall process variation. For processes with two-sided specification limits, the percentage of nonconforming items (%NC) can he calculated as I - F(USL) + F(LSL), where F(.) is the cumulative distribution function of the process characteristic X. On the assumption of normality, %NC can he expressed as:

USL - p LSL - p %NC= I - o ( ~ ~ + o ( ~ ) ,

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PROCESS ACCURACY AND PRECISION INDICES 987

where a( . ) is the cumulative function of the standard normal distribution. If the process is perfectly centered, then %NC can be expressed alternatively as %NC = 2 - 20(3q) . For example, C, = 1.00 corresponds to %NC = 2700 ppm, and C, =

1.33 corresponds to &NC = 63 ppm. Thus, the index C, provides an exact measure of the actual process yield. Since C, measures the magnitude of process variation. C, may be viewed as aprocess precision index.

While the precision index C, measures the magnitude of process variation, the index k measures the departure of process mean, p, from the center-point m. Therefore, the transformation of k defined as C, = 1 - k measures the degree of process centering (the ability to cluster around the center), which can be regarded as a process accuracy index. For example, C, = 1 indicates that the process is perfectly centered (p = m), C, > 1/ 2 indicates that p is within half of the specification interval, and C, = 0 indicates that p is on the specification limits (c( = USL, or p = LSL). On the other hand. if C, c 0 then it indicates that p falls outside the specification limits (p > USL, or p < LSL). Obviously, the process is severely off-center and it needs an immediate troubleshooting.

2. ESTIMATION OF C ,

To estimate the precision index C,, we consider the natural estimator ?, defined as the following, where S = [EL1 (Xi - Z)'/(n - 1)]In is the conventional estimator of the process standard deviation o, which may be obtained from a stable process.

The natural estimator ?, can be al[ernatively written as:

On the assumption of normality, the statistic (n - 1)S2/& is distributed as Xi . I , a chi-square with n - 1 degrees of freedom. Therefore, the probability density function of ?, can be expressed as (Chou and Owen (1989)):

for x > 0. The r-th moment of c,, therefore can be calculated as the following:

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988 PEARN. LIN. AND CHEN

and the first two moments as well as the variance nay be obtained as (see also

Chou and Owen (1989). and Pearn, Kotz and Johnson (1992)):

It can be shown that the coefficient of ~ ( e , ) , [(n - 1)/2]lR T[(n - 2)/2]/r[(n - 1)/2] > 1 for all n. For n 2 15, this coefficient can be accurately approximated by (4n - 4)/(4n - 7). Therefore, the natural estimator e, is biased, which over- estimates the actual value of C,. Table 1 displays the values of E(G) under the condition C, = 1 for various sample sizes n. For the percentage bias to be less than one percent ( I E ( ~ , ) - CpI/Cp 10.01). it requires the sample size n > 80.

- we may obtain an unbiased estimator C, = b,&. That is, ~ ( e , ) = C,. Since br < 1, then the variance of e, is smaller than that of the natural estimator ep. That is, var(?,) < var(Ep). In the following, we investigate the statistical properties of e,. We show that e, is the UMVUE of C,, which is consistent and asymptotically efficient.

An estimator in of e is said to be consistent if for all e > 0, - 01 > E) + 0 A

as n -+ m for all 8. A sufficient condition for the consistency is that E(e,) + e and

~ a r ( i , ) -+ 0. Under regular conditions, the estimator 8, is said to be A A ,.

asymptotically efficient if 8, is asymptotically normal, nin [en - E(0J + 0, and {n

~ a r [ i , - E(;j,)]) converges to the Cramer-Rao Bound.

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PROCESS ACCURACY AND PRECISlON INDICES

Table 1. Valucs of E(?,) corresponding to C, = 1 for various sample sizes n.

Theorem 1. If the process characteristic follows normal distribution, then

(a) is the UMVUE of C,. (b) is consistent. (c) nIn (?, - C,) converges to N(0, [CPl2/2) in distribution. (d) e, is asymptotically efficient.

Proof: (a) We first note that the statistic R, S2) is sufficient and complete for (p, oZ). Since E(?,) = C,, and e, is a function of O(. S2) only, then by Lehmann-Scheffe' Theorem (Arnold (1990)) e, is the UMVUE of C,.

(b) For all E > 0, p(i?p - C,I > E) < E(?, - Cp)?/@. NOW, E(?, - Cp)2 = ~ a r ( G ) = E(?,)z - C,Z. By Stirling's formula, we can show that E(E,)~ converges to C,?. Hence, E(?, - Cp)2 converges to zero. Therefore, ?, converges to C, in probability and ?, must be consistent.

(c) If the process characteristic is normally distributed, then it is clear that the statistic nln(S2 - 02) converges to N(0, 204) in distribution. We apply Cramer-8 Theorem (Arnold (1990)) with g(t) defined as g(t) = d/(3t1n). Since gt(t) = - d/(6t3"), and [g'(o*)]2 = (CP)2/(4o4), then n'" [g(S2) - g(a*)] = nln [d/(3S) - d/(30)] = nl"(& - C,) converges to N(0, 204[g'(02)]?), or N(0, [Cp]2/2) in distribution. By (b) ?, converges to C, in probability, then by Slutzky's Theorcm (A~mold (1990)) nln(ep - ep) converges to N(0, [Cp]2/2), and so nll?(ep - C,) converges to N(0. [CPl2/2) in distribution.

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994 PEARN, LIN, AND CHEN

If the knowledge on whether p(p 2 m) = 1, or 0 is available, then we can - consider the estimator C, = 1 - [(X - m) sgn(p - m)]/d, where sgn(p - m) = 1 if p - m 2 0, and sgn(p - m) = -1 if p - m < 0. Thus, c, = 1 - (X - m)/d if p 2 m, and e, = 1 - (m - X)/d if p < m. We can show that the estimator Fa is the MLE, and the UMVUE of C,. We can also show that the estimator c, is consistent, and efficient.

Theorem 3. If the process characteristic follows normal distribution, then

(a) E, is the MLE of C,. (b) c, is the UMVUE of C,. (c) Z', is consistent. (d) nl" (c, - C,) converges to N(0, 1 /[3Cp]2) in distribution.

Proof: (a) We first note that the statistic (?, [(n - l ) /n] S2) is the MLE of (F, oZ). By the invariance property of the MLE, is the MLE of k, and Fa is the MLE of C,.

(b) From Theorem 2(d), the Cramer-Rao bound = 1/[9n Cp2]. Since e, is distributed as N(C,, 1/[9nCpZ]), then ?, is efficient for C,, and is the UMVUE of C,.

(c) For all E > 0, p(lea - CaI > E) < E(E, - Ca)2/~2. Now, E(?, - Ca)2 = ~ a r ( P , ) = l/[9nCp2] converges to zero, and so Z', must be consistent.

(d) From (c), c, converges to C, in probability. From Theorem 2(c), niR (Fa - C,) converges to N(O, 1 /[3Cp]2) in distribution. By Slutzky's Theorem (Arnold (1990)). niR (c, - C,) converges to N(0, 1 /[3CPl2) in distribution..

In fact, since and S' are mutually independent, thcn ?, and c, are also mutually independent. Therefore, since Z = 3 f i cC,(ea - c,) is distributed as N(0, 1) and W = (n - 1)(6,~,~/[?,]2 is distributed as xi-I, then 3ni~?,(C, - C,)/bt = Z/[W/(n-l)]lR is distributed as r,.,, a t distribution with n - 1 degrees of freedom. Therefore, the a-level confidence interval for C, can be established as:

where tn.l,, is the upper a-th quantile of the r n . ~ distribution. The length, l', of

the confidence interval, therefore is

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PROCESS ACCURACY AND PRECISION INDICES

Table 4. E(1' ) for C, with C, = 1 and a = 0.05

Sample Size E(1') Sample Size E(1')

The expected value E(I' ) and the variance Vat(l' ) of the length of the confidence interval I' , can be found as:

for given sample size n. Table 4 displays the expected lengths, E(1' ), of the a- level confidence intervals for the accuracy index C, with Cp = 1 and a = 0.05 for various sample sizes n.

4. TESTS FOR PROCESS CAPABILITY

To judge whether a given process meets the preset capability requirement and runs under the desired quality condition. We can consider the following statistical testing hypothesis for C,: Ho: C, S C, and H I : C, > C. Process fails to meet the capability (quality) requirement i f C, 5 C, and meets the capability

requirement if C, > C. We define the test @"(x) as: @*(x) = 1 if e, > co , and $*(x) = 0 otherwise. Thus, the test $* rejects the null hypothesis Ho (C, < C ) if 2, > Q,

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PEARN. L I N . A N D CHEN

Tahle 5. Critical valucs co for C = 1.00 with n = 10(10)100, and a = 0.01,0.025,0.05.

-----..----------.--...--------------.---------------.-----

Sample size 0.01 0.025 0.05

with type I error a(co) = a, the chance of incorrectly judging an incapable process (C, 5 C) as capable (C, > C). The critical value, CO, can be determined as:

Hence, we have

whcre xA-1, a is the lower a-th quantile of xi., distribution, or,

Therefore, if ?, > CO, then $*(x) = 1 and we reject the null hypothesis Ho and conclude that the process meets the capahility requirement (C, > C). Otherwise, we can not conclude that the process meets the capability requirement. Tables 5 displays the critical values co for C = 1.00 with sample sizes n = 10(10)100, and a- risk = 0.01.0.025, 0.05 (the chance of incorrectly concluding a process with C, I C as one with C , > C).

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PROCESS ACCURACY AND PRECISION INDICES 95'7

Theorem 4. For the testing hypothesis Ho: C, I C and HI: C, > C, the test defined as $*(x) = 1 if e, > CU, and $*(x) = 0 otherwise, is the UMP test of level a, where c, is determined by E,[~*(x)] = a..

Proof: For the test, the power function is:

FOP a(co) = a , co = brm where 1-0 satisfies G'

Since for C; > C, > 0, B(x'* C; ) > f d x v if and only if > x > 0, f~p(x', c p ) f6(% cp )

then { f ~ ~ (x, C, ) I Cp > 0 ) has MLR (monotone likelihood ratio) property in c,. Therefore, the test 4' must be the UMP test.

5. CONCLUSIONS

Process capability indices such as C,, k, and Cpk, have been widely used in manufacturing industry to provide numerical measures on process potential and performance. The index C, measures the overall process variation, and the index k measures the degree of process departure. In this paper, we considered C, and a transformation of k defined as C, = 1 - k. We referred to C, as the process precision index, and C, as the process accuracy index which measures the degree of process centering.

We considered the estimators of C, and C,, and investigated their statistical properties. For C,, we obtained the UMVUE and the MLE. We showed that this UMVUE is consistcnt, and asymptotically efficient. For C,, we investigated its natural estimator. We showed that this natural estimator is the MLE, which is asymptotically unbiased and asymptotically efficient. In addition, we proposed an efficient test based on the UMVUE of C,. Using this test, the practitioners can judge whether their processes meet the capability requirement preset in the factory. We showed that the proposed test is in fact the UMP test.

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998 PEARN, LIN, A N D CHEN

Appendix 1

Theorem 1: The probability density function of ?, can he expressed as:

Proof: For - - < x l 1, the probability density function f(x) is

d A d A .. ,. f(x)=-p(C,Ix)=-p(1 - k I x ) = d - p ( k 2 1 - x ) = d - ( 1 - p ( k I 1 - x ) }

dx dx dx dx

= g(1 - x), where g(x) is the probability density function ofc.

- Now. the statistic Y =- is distributed as N (y . [9n ( ~ , ) 2 ] ' 1 .

d a normal distribution with mean py = (p - myd and variance = [9n (cPf2]".

Since = Ill = has a folded normal distribution, then the probability

density function of is:

= 6 C, cosh ( 9 n [C,]' k y ) exp I Therefore, the probability density function f(x) is

f ( x ) = 6 ~ , ~ c o s h ( 9 n [ ~ ' k ( l x - X I ) exp[ - 9 n ( ~ , f [ ( l - x f + k 2 ] 2

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PROCESS ACCURACY AND PRECISION INDICES

Appendix 2

Theorem 2: The first two moments of ea are:

Proof: For simplicity of the derivation of the exact formulae for the moments, we assume that p 2 m. For the other case, p < m, the derivation and the result will be the same. From Theorem 1, the probability density function of is

Therefore.

E [ C I = ~ O ~ 7E exp

1 exp - (c,)Y] + t [ I - 2 a (- 3 f i k cP)] =G [ 2

1 = 9n(Cp)?(1 - Ca)2 + 1 - 9 n ( c p - Cpr)'+ 1 = ( 1 -cay+---- -

9 I, G)z 9 n (c,)? 9 n (CPY

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PEARN, LIN. A N D CHEN

Then, we may obtain E[(C,) and E[(c,)?] as:

BIBLIOGRAPHY

1. Arnold, S. F. (1990). Mathcrnatical Statistics. Prentice Hall.

2. Chou, Y. M. and Owen, D. B. (1989). On the distributions of the estimated process capability indices, Cornmunicariorzs in Statistics - Theory and Methods, 18, (12). 4549-4560.

3. Kane, V. E. (1986). Process capability indices, Journal of Q~rality Technolo~y, 18, (I), 41-52.

4. Leone, F. C., Nelson, L. S. and Nattingham. R. B. (1961). The folded normal distribu~ion, Techlmrwrrics, 3, (4), 543-550.

5. Peam, W. L., Kotz, S. and Johnson, N. L. (1992). Distributional and inferential properties of process capability indices, Journal of Quuliv Techrwlo~y, 24, (4), 216-233.

Received May, 1997 ; Revised October, 1997.

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